# A Mathematical Dilemma

• 22 months ago · Quote · #61
• 22 months ago · Quote · #62

Pie is better than pi. Just sayin...

• 22 months ago · Quote · #63

0 times infinity is useful as an "indeterminate form" in the study of limits, in the mathematical basis of differential and integral calculus, but I don't think it has anything to do with chess and definitely not with the post that began the thread unless you want to consider an infinite chessboard with infinitely many chess pieces on it.  Any particular game would take place in a finite area, but the union of all the games would take up the entire infinite board (I'm thinking it would be 8 ranks and infinitely many files.)

• 22 months ago · Quote · #64

Take the limit of Houdini's eval function as knight --> captured = ???

On second thought lets not go to chess-calculus, tis a silly place

• 22 months ago · Quote · #65

making a fraction with a zero always leads to nonsense. The first analyses is correct

• 22 months ago · Quote · #66

If we want to be accurate then in the starting setup the loss ratio would be 27:34 (assuming that king doesn't have value). After the takes the loss ratio would be 30:44.

So it's still better to take the rook.

• 22 months ago · Quote · #67
paul211 wrote:
FTLulz wrote:

You'd think that even with a BA in math that paul would know the correct value of pi.  Of course, maybe in the "system" (or language?) he is using, his value is correct.

LESS WE FORGET the M.I.T. fight song:

SIN - SIN - COSINE - SIN

3 - POINT - 1 - 4 - 1 - 5 - 9

GOOOOOOOO TEAM!

Would not mind meeting the co-sine!

The value of π, pi and not pie is universal and transcends languages and I think that you know this but not sure when reading your intervention.

The value of pi is approximately 3.1415926535 or so I used to know 128 numbers but I forget now.

Unless one is using it at a galactical level the the approximation holds.

Ah, nice to know that we agree on the value of pi being universal, especially given that you claimed the value to be 3.141516 in an earlier post (to point out the obvious, the last two digits you gave were incorrect).  Possibly I was too subtle in my previous "intervention" for you to realize the point of my post.

• 22 months ago · Quote · #68

The mathematics is good enough, but the economics is faulty.  No account is taken of opportunity cost: what havoc Black could have inflicted on white had he not spent that second rook, R2, avenging the loss of his first rook, R1?

• 21 months ago · Quote · #69

Initially the 5:0 ratio is different case from the 10:3 ratio.

In the first case the 5:0 ratio means you lose 5 points to the opponent losing zero.But 5:0 ratio is also equal to 500:0 or as such anything is to zero because it is a fraction in the basic terms.But here it means you are losing 500 points to the opponent losing only zero(however absurd it sounds).

But in the other case it changes.Here 10:3 may be equal to either 20:6 or any subsequent product of it.

So you can just use ratios for relative measure.Not a quantative measure

• 21 months ago · Quote · #70
pavankumartgpk wrote:

Initially the 5:0 ratio is different case from the 10:3 ratio.

In the first case the 5:0 ratio means you lose 5 points to the opponent losing zero.But 5:0 ratio is also equal to 500:0 or as such anything is to zero because it is a fraction in the basic terms.But here it means you are losing 500 points to the opponent losing only zero(however absurd it sounds).

But in the other case it changes.Here 10:3 may be equal to either 20:6 or any subsequent product of it.

So you can just use ratios for relative measure.Not a quantative measure

And just as an afterthought anything divided by zero is not infinity.It limits to infinity

• 21 months ago · Quote · #71

Some have commented that ratios may be a viable way to assess the values of exchanges.  It can be shown that:

Case 1:  Ratio of exchanged pieces à hahaha, um, nonsense

Case 2:  Ratio of pieces remaining on board  à leads to the same decision as the simple difference of exchanged pieces

For the purposes of this exercise, we will ignore all other considerations except for the unrealistic assigning of unchanging values to every piece (yes, even the King).  This is known as – and then the physicist said, “First, assume that the horse is a sphere” - approximation.

Assignment of Variables:

A = White’s material before exchange         B = Black’s material before exchange

C = Whites’ loss of material due to exchange (for example, if White loses two pawns, C  = +2)

D = Black’s loss of material due to exchange (for example, if Black loses three pawns, D  = +3)

To reduce the confusion, it may be of help to look at this from, for example, White’s perspective.

C < D is “Good”  and   C > D is “Bad”   Thus, the conventional simple difference of exchanged pieces (D – C) leads to  (D – C) > 0 is “Good”  and   (D – C) < 0 is “Bad” from White’s perspective

I will move straight to Case 2 because, well, Case 1 has already been explained.  I will reiterate Case 1 for the purposes of completeness and laughs at the end.

Case 2:

Before the exchange the situation is White has A and Black has B material.  After the exchange White has (A – C) and Black has (B – D) material remaining.  We can now ask “How does the ratio of White’s material to Black’s material after the exchange compare with the ratio before the exchange.   We can take the difference of these two ratios to see if the ratio of White’s material to Black’s has increased.  Some have intuitively and correctly mentioned that they believe an increase will represent a “Good” exchange for White (yea, yea, the horse, remember?).  Thus a Difference of Ratios > 0 is “Good” for White etc.

Material ratio White to Black before exchange = A / B

Material ratio White to Black after exchange = (A – C)  / (B – D)

Difference of Ratios = { (A – C)  / (B – D) }  -  { A / B }     ; finding a common denominator (cheaper than the uncommon, rare or epic denominators) etc. we arrive at

Difference of Ratios = { (D – C)  / (B – D) }         Hmmm, interesting thing about A, B, (A – C) and (B – D), they are all positive quantities.

Thus the Difference of Ratios HAS THE SAME SIGN (i.e. “Good” or “Bad”) as the simple difference of exchanged pieces that is (D – C). To evaluate two exchange scenarios, consider one as the initial condition blah, blah, blah.

Since I do not have a BA in math, you can assume this to be “gospel”.  JUST KIDDING PAUL!

Case 1:

We will eliminate the use of the ratio of exchanged pieces by the invoking of an enlightening example.

Allow A and B to be “big” (so that we can have “big” C’s and D’s).

Case1A:  C = 2, D = 1   a Loss Ratio of White to Black = 2

Case1B:  C = 30, D = 16   a Loss Ratio of White to Black < 2

As suggested in the OP, we are initially assuming that a smaller Loss Ratio would be desired.  What we find is that in the supposedly undesirable scenario, Case1A, we (White) lose one extra pawn, in the preferred scenario, Case 1B, we lose our @#\$.

GGG - (“Good Game Geek”)

• 21 months ago · Quote · #72

@paul211

Ok, so its agreed that joking on my part about memory is off-limits. I will assume an effort on your part to see the humor in what is otherwise posted.

• 21 months ago · Quote · #73

I would like some math courses :D

• 21 months ago · Quote · #74
FTLulz wrote:

Some have commented that ratios may be a viable way to assess the values of exchanges.  It can be shown that:

Case 1:  Ratio of exchanged pieces à hahaha, um, nonsense

Case 2:  Ratio of pieces remaining on board  à leads to the same decision as the simple difference of exchanged pieces

For the purposes of this exercise, we will ignore all other considerations except for the unrealistic assigning of unchanging values to every piece (yes, even the King).  This is known as – and then the physicist said, “First, assume that the horse is a sphere” - approximation.

Assignment of Variables:

A = White’s material before exchange         B = Black’s material before exchange

C = Whites’ loss of material due to exchange (for example, if White loses two pawns, C  = +2)

D = Black’s loss of material due to exchange (for example, if Black loses three pawns, D  = +3)

To reduce the confusion, it may be of help to look at this from, for example, White’s perspective.

C < D is “Good”  and   C > D is “Bad”   Thus, the conventional simple difference of exchanged pieces (D – C) leads to  (D – C) > 0 is “Good”  and   (D – C) < 0 is “Bad” from White’s perspective

I will move straight to Case 2 because, well, Case 1 has already been explained.  I will reiterate Case 1 for the purposes of completeness and laughs at the end.

Case 2:

Before the exchange the situation is White has A and Black has B material.  After the exchange White has (A – C) and Black has (B – D) material remaining.  We can now ask “How does the ratio of White’s material to Black’s material after the exchange compare with the ratio before the exchange.   We can take the difference of these two ratios to see if the ratio of White’s material to Black’s has increased.  Some have intuitively and correctly mentioned that they believe an increase will represent a “Good” exchange for White (yea, yea, the horse, remember?).  Thus a Difference of Ratios > 0 is “Good” for White etc.

Material ratio White to Black before exchange = A / B

Material ratio White to Black after exchange = (A – C)  / (B – D)

Difference of Ratios = { (A – C)  / (B – D) }  -  { A / B }     ; finding a common denominator (cheaper than the uncommon, rare or epic denominators) etc. we arrive at

Difference of Ratios = { (D – C)  / (B – D) }         Hmmm, interesting thing about A, B, (A – C) and (B – D), they are all positive quantities.

Thus the Difference of Ratios HAS THE SAME SIGN (i.e. “Good” or “Bad”) as the simple difference of exchanged pieces that is (D – C). To evaluate two exchange scenarios, consider one as the initial condition blah, blah, blah.

Unfortunately, {(A - C)/(B - D)} - {A/B} = {[AD - BC]/[B(B - D)]}, which is NOT equal to     {(D - C)/(B - D)} unless it is assumed that A = B. No such assumption is stated, however. In other words, White and Black need to have started with the same amount of material in order to guarantee that the difference between ratios of remaining pieces will have the same sign as the difference of exchanged pieces.

• 21 months ago · Quote · #75

@cobra91

How dare you defame the "gospel"?  good story tho, bro.

Yes, you are right, the horse is not a sphere.  I knew I should have went with at least the uncommon denominator.

• 21 months ago · Quote · #76
RainbowRising wrote:

Your analysis fails on the grounds that your ratio should be the strength of the pieces left on the board, not the ones taken off.

I thought the same thing, this is the most appropriate answer IMO.

• 21 months ago · Quote · #77

I have seen this before. I was playing someone some years ago who was obviously struggling quite a bit. I said he'd be a piece down after a capture of mine. He denied it, took an unrelated bishop of mine and I took back. Though I had made two captures and he had made only one, he didn't think he was a piece down! So this type of thinking should always be avoided. Just look what is on the board and check what you have left after the exchange. It's that simple.

• 20 months ago · Quote · #78
stromy_king wrote:

Although the part of brain used while playing Chess is same as that of Maths.

The hippocampus... Yeah!

• 20 months ago · Quote · #79

It's the part of the brain used for evaluation and judgement, if I remember correctly.

Correct me if I'm wrong.

• 20 months ago · Quote · #80

I would say the main problem with a ratio method of evaluation is that all ratios are equal ie.  a 6:3 advantage is the same as a 2:1 advantage.  Of course winning a rook and pawn for three pawns is much better than just winning a pawn.